Problem 26

../

claims/: The 3 claim pages of Problem 26, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$?

Status. DISPROVED (LEAN).

Source. erdosproblems.com/26, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #26, https://www.erdosproblems.com/26.

References.

• [DaEr51] Davenport, H. and Erdős, P., On sequences of positive integers. J. Indian Math. Soc. (N.S.) (1951), 19-24.
• [Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
• [Te13] Tenenbaum, Gérald, Some of Erdős' unconventional problems in number theory, thirty-four years later. (2013), 651-681.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation (accessed 2026-09-04; the site's page was last edited 2026-04-08) asks whether every infinite $A\subset\mathbb{N}$ has a shift $k$ for which almost all integers have a divisor in $A+k$. The answer is no, and the standing derives from two accepted full claims: Davenport and Erdős 1951, whose theorem that a set of multiples of density one needs a divergent reciprocal sum predates the question and refutes it for every infinite set with a convergent reciprocal sum, and Ruzsa's counterexample, an explicit sequence built by the Chinese remainder theorem whose shifted copies all miss a positive-density progression, whose existence Erdős reported in [Er95]. Both are accepted on the site curator's credit; the 1951 paper is refereed. The site reports van Doorn's modification of Ruzsa's construction to a set with divergent reciprocal sum, and the formal-conjectures statement file restricts the question to such sets while the site's statement does not.

The site's label carries a Lean qualification: the community database records that the statement and its resolution are both formalized, the resolution in Boris Alexeev's repository as a formalization of Ruzsa's construction, linked from that claim page. The file proves Ruzsa's set as its own statement and the formal-conjectures variant for sets with convergent reciprocal sum with the witness $2^{2^n}$; the project's main statement, which restricts the question to sets with divergent reciprocal sum, is not proved by it, although the statement file points to it. The file is third-party Lean that this corpus has not built, so no claim lists formalized evidence.

Tenenbaum's weaker variant, that for every $\epsilon>0$ some shift makes the multiples of $A+k$ reach density at least $1-\epsilon$, is a different question. The site reports it resolved in the negative by a DeepMind prover agent, with an infinite $A$ such that for every $k\geq 1$ the multiples of $A+k$ have upper density below $0.34$; the result was posted in the site's thread on 2026-04-06 with a Lean proof in a fork of formal-conjectures. It implies the negative answer here as well, with a set of divergent reciprocal sum, so it also refutes the formal-conjectures statement restricted to such sets; it is recorded on its claim page, which stays claimed, since the curator credits the result for the variant only and this corpus has not built the file.

Search scope: on 2026-10-07 the site's problem page and thread, the community database (teorth/erdosproblems, data/problems.yaml), the formal-conjectures statement file and the Lean repository named above were read; no further claim on the stated question was found. Nothing remains open in the stated question.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• davenport_1951_sequences_positive_integers
• davenport_1951_sequences_positive_integers / remark_p19
• tenenbaum_2013_erdos_unconventional_problems_number_theory
• tenenbaum_2013_erdos_unconventional_problems_number_theory / equation_27